First slide

Theory of equations

Question

If at least one root of $2 x^{2} + 3 x + 5 = 0$ and ${ax}^{2} + bx + c = 0, a, b, c \in N$ is common, then the maximum value of a+ b +c is

Moderate

A

10
B

0
C

does not exist
D

None of these

Solution

Roots of the equation $2 x^{2} + 3 x + 5 = 0$ are
$x = \frac{- 3 \pm \sqrt{9 - 40}}{6}$ (imaginary roots)
Hence, both roots coincide, so on comparing
$\frac{a}{2} = \frac{b}{3} = \frac{c}{5} = k \begin{array}{lrl} \Rightarrow a = 2 k, b = 3 k, c = 5 k \\ \Rightarrow a + b + c = 10 k \end{array}$
So, maximum value does not exist.

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